A wide and growing range of different types of derivatives is available in financial markets around the world. Broadly speaking, a derivative is an option to buy or sell an underlying asset in the future at a given exercise price, which may be fixed in advance or determined by conditions yet to occur. The asset typically comprises a financial instrument, such as a share, collection of shares or bonds traded on a public market. There are also derivatives based on other assets or instruments, such as commodities, currency values, share price indices and even other derivatives. In the context of the present patent application and in the claims, the term “asset” refers to any and all such entities that may underlie a given derivative. The challenge for the trader in derivatives is to determine, based on the price and expected behavior of the underlying asset, whether the current purchase price of the derivative is justified by the profit that is likely to be made from its future exercise and the potential risk.
The simplest type of derivative is known as a European “vanilla” option. It gives the buyer the right to buy (“call”) or sell (“put”) the underlying asset at a certain fixed time in the future at a given exercise price. The buyer should purchase the option if the expected difference between the exercise price and the actual price of the asset at the specified time of exercise is greater than the current price of the option while the risk is within defined limits. American options allow the buyer to buy or sell the asset at the specified price at any time in the future, typically up to a certain fixed time limit. In this case, the buyer must decide both whether the current option price is justified and, after buying the option, when is the best time to exercise it.
While traditional European and American vanilla options specify a fixed exercise price and conditions, path-dependent options can have variable exercise prices and conditions, which depend on the behavior of the underlying asset over time. Such exotic options are described at greater length by Hull, in Options, Futures and Other Derivatives (Prentice Hall, 1977), pages 457-489, which is incorporated herein by reference. In Asian options, for example, the exercise price is based on an average price of the underlying asset, taken over a certain period prior to exercise. Another example of a path-dependent option is the barrier option, which becomes exercisable only when the price of the underlying instrument rises above or falls below a given threshold.
Even the simplest vanilla options pose a substantial challenge to the derivatives trader, because of the volatile nature of share price values in the stock markets. In deciding whether and when to buy, sell or exercise a given derivative, the trader must assess the trends driving the price of the underlying instrument and the likely variability of the price. These assessments must be factored into a model that can form the basis of a trading strategy that maximizes the profitability of the investment. Models used for this purpose typically view the asset price as a stochastic process.
Various methods are known in the art for determining the stochastic behavior of a process (S) over time (t), based on a trend function μ(S,t) and a variance function σ(S,t). In the simplest case, S is a scalar, representing the price of a given asset over time. This simplistic model does not reflect the complex reality of financial markets, however. More generally, S is a vector that includes a number of related economic or financial factors, so that μ(S,t) is similarly a vector, and σ(S,t) is a covariance matrix. The change in S over time is modeled as a multidimensional Wiener process, given by:dS=μ(s,t)*dt+σ(s,t)*dW  (1)Here dW is a multidimensional Wiener differential, wherein each component of dW is distributed randomly with a normal probability density. This model is well known in the art and is described at greater length in the above-mentioned book by Hull, for example.
In order to evaluate a financial derivative based on S, it is necessary to solve equation (1), based on specified boundary conditions. Black and Scholes showed that when S can be treated as a scalar log-normal process, equation (1) can be used to derive a differential equation giving the price of a derivative contingent on S. The derivation of the Black-Scholes equation and its use in analyzing options are described by Hull on pages 229-260. There are many derivatives, however, to which the Black-Scholes analysis is not applicable, such as path-dependent and other exotic options and, more generally, processes that cannot be properly modeled as log-normal.
For these more complex analyses, a number of alternatives are known in the art, for example:                Monte Carlo methods can be used to simulate the behavior of the underlying asset and/or derivative over time. Such simulation becomes extremely computation-intensive, however, when high accuracy is required, and small values of the time step dt must be used. Furthermore, Monte Carlo methods are not applicable to American-style path-dependent options.        Numerical solutions of partial differential equations can be used to evaluate some types of derivatives, but they are very sensitive to choice of boundary conditions and are likewise demanding of computation resources.        Binary and tertiary trees are useful particularly in some path-dependent assessments, but are limited in their ability to deal with multi-dimensional problems. They also require a great many nodes of the tree to be evaluated in order to achieve high accuracy.A further limitation of all of these methods is that they typically provide only the expected value of the asset or derivative at a given time, without indicating the probability distribution of possible values about this expectation. This distribution can be very important in determining the risks associated with a given derivative.        